Cohomology and Metrics of Tensor Spaces

Authors: Matej Dolezalek, Nikhil Ken

Cohomology of Tensor Spaces

The study of equations of secant varieties involves the search of semicontinuous invariants. One family of methods consists of interpretting the tensor as a space of matrices, embedding it in a compactification with well-understood cohomology and determining the class of the resulting variety. In the context of algebraic geometry, we realize the cohomology ring as the Chow ring. Any intersection number with then behave semicontinuously.

The theory of matroid geometry is a useful blueprint. A representable matroid may be treated as a subspace $L\subset {\mathbb{C}}^n$, where ${\mathbb{C}}^n$ is considered not merely as a vector space, but as a vector space with a distinguished basis. An equivalent way to achieve this is to identify it with diagonal matrices. The following constructions have been well-studied:

• The matroidal Schubert variety (or arrangement Schubert variety) $Y_L$ obtained by compactifying $\mathrm{diag}(n)\hookrightarrow ({\mathbb{P}}^1{)}^n$.
• The wonderful compactification $W_L$ obtained by taking a Chow quotient of $({\mathbb{P}}^1{)}^n$ with respect to a ${\mathbb{C}}^{\ast }$-action, yielding the permutohedral variety $\mathrm{Perm}(n)\supset W_L$.

Analogous steps may be taken for tensors: a (concise) tensor is represented as a space of matrices, these are then compactified in the Grassmannian $\mathrm{Gr}(n,2n)$, yielding the generalized Schubert variety $Y_L$, also called a Schubert compactification of the linear space $L$. A Chow quotient of $\mathrm{Gr}(n,2n)$ then yields the variety of complete collineations $\mathcal{CC}(n)$. The description of the Chow ring of $\mathrm{Gr}(n,2n)$ is classical, with an additive basis given by Young diagrams and multiplication being governed by the Littlewood-Richardson rule, all truncated to the $n\times n$ square. The Chow ring of complete collineations is less understood, though the subring generated by divisors (hyperplane pullbacks under the Plücker embedding) has been studied.

As an easy example, for a $1\times n\times n$ tensor $T$, which we may identify with the $n\times n$ matrix generating it as a one-dimensional space, the intersection number with the hyperplane class in $\mathrm{Gr}(n,2n)$ recovers $\mathrm{deg}([Y_T]\cdot \square )=\mathrm{rank}T$.

Similarly, partially symmetric tensors, giving subspace of symmetric matrices ${\mathrm{Sym}}^2{\mathbb{C}}^n$, can be compactified in the Lagrangian Grassmannian $\mathrm{LG}(n)$, which may then be quotiented to obtain the variety of complete quadrics $\mathcal{C}\mathcal{Q}(n)$, itself a classical object. The Chow ring of $\mathrm{LG}(n)$ is also classically described, though somewhat unwieldy — a basis is given by Young diagrams with nonrepeating rows, and a complicated analogue of the Littlewood-Richardson rule holds.

Overall, the compactifications and quotients described above may be neatly summed up in the following diagram:

Nonlinear flattenings

Homological techniques are also connected to flattenings methods. A flattening map associated polynomially a matrix to a tensor, while respecting the actions of $\mathrm{GL}$ groups. This assures that the rank of flattening is a semicontinuous function of the tensor. Beyond well-studied linear flattenings such as Koszul and Young flattenings, we have explored nonlinear methods, dubbed Kronecker-Koszul flattenings.

These take a tensor power of a tensor, tensor it futher with various identity maps and apply a suitable representation-theoretic contraction, and finally flatten classically. This has several advantages over linear methods: it avoid the so-called cactus barrier, while also allowing a “structured” access to high-degree equations.

A very fruitful example of a Kronecker-Koszul flattening is the tangency flattening. To a tensor $T\in A\otimes B\otimes C$, it associates the matrix

$${\Phi }^{\mathrm{tg}}(T):\stackrel{3}{\bigwedge }{A}^{\vee }\otimes B^{\vee }\otimes C^{\vee }\to A^{\vee }\otimes C\otimes B,$$ $$({\alpha }_1\wedge {\alpha }_2\wedge {\alpha }_3)\otimes \beta \otimes \gamma \mapsto \sum _{\sigma \in {\mathfrak{S}}_3}\mathrm{sgn}(\sigma ){\alpha }_{\sigma (1)}\otimes T({\alpha }_{\sigma (2)},\beta )\otimes T({\alpha }_{\sigma (3)},\gamma ).$$

If $R$ is an $n$-dimensional commutative unital algebra, ${\mu }_R\in R^{\vee }\otimes R^{\vee }\otimes R^{\vee }$ its structure tensor and $S=\mathbb{C}[x_1,\dots ,x_d]$ a polynomial algebra with a surjective ring morphism $S\twoheadrightarrow R$, then the tangency flattening relates to the tangent space to the Hilbert scheme of points via

$$\mathrm{rk}{\Phi }^{\mathrm{tg}}({\mu }_R)-n(n-1)(n-2)=dn-\dim T_{[S\twoheadrightarrow R]}{\mathrm{Hilb}}_n(S)=dn-\dim {\mathrm{Ext}}_1^S(R,R).$$

In the expression, the tangent space is identified with the Ext module. A generalization may be obtained for ${\mu }_M$ a structure tensor of a finite-dimensional $S$-module.

Relevant articles: arXiv:2602.12762, arXiv:2610.17875

TENORS talk slides

Metrics of Tensor Spaces

Tensor product of Hilbert spaces admits a natural metric which allows us to study optimization problems.

In the Euclidean setting, we work with a nondegenerate symmetric bilinear form $q$ and consider

$$d_u(x)=q(u-x,u-x).$$

The critical-point condition is

$$q(u-x,v)=0\text{for every }v\in T_{x}X.$$

For generic $u$, the number of complex critical points is finite and constant, and is called the Euclidean distance degree, denoted $\mathrm{EDdeg}(X)$.

For the squared Hermitian distance

$$h_u(x)=⟨u-x,u-x⟩,$$

the corresponding condition is Hermitian orthogonality:

$$⟨u-x,v⟩=0\text{for every }v\in T_{x}X.$$

These equations involve complex conjugation and hence define a real algebraic critical-point problem rather than a holomorphic one. The number of critical points can vary between chambers in the space of data tensors. This motivates the Hermitian distance degree (HD degree) as a real enumerative invariant.

A complementary algebraic count is obtained by complexifying this real algebraic system: variables and their complex conjugates are replaced by independent variables. The generic number of solutions of the resulting doubled system is the virtual Hermitian distance degree (vHD degree). It provides an algebraic upper bound for the number of isolated Hermitian critical points for generic data.

We study these invariants for structured varieties, including unitary-invariant matrix varieties and the Veronese cone of symmetric rank-one tensors. If $M\subset {\mathbb{C}}^{m\times n}$ is invariant under the action of $U(m)\times U(n)$, its Hermitian critical-point problem reduces to a Euclidean critical-point problem on an associated absolutely symmetric singular-value slice $S\subset {\mathbb{R}}^{\min (m,n)}$. At the level of enumerative invariants, this yields

$$\mathrm{HDdeg}(M)=\mathrm{rEDdeg}(S),$$

where $\mathrm{rEDdeg}$ denotes the real Euclidean distance degree.

Relevant article: Nikhil Ken, Hermitian Distance Degree of Unitary-Invariant Matrix Varieties, arXiv:2602.10737.

Courses

• Scalar products and metric considerations on tensor spaces Giorgio Ottaviani (UNIFI) - Hand-out Part 1 Hand-out Part 2 Exercises - Recording (Part 1) Recording (Part 2)